This problem should have been posted with a diagram, because some of the points (D,F) are undefined. Assumptions have therefore been made to make it a sensible question. So please VERIFY diagram with your given diagram to confirm the question BEFORE reading further.
We are given EA and EC are common tangents to circles G & B.
(assumed mEG < mEB, and A, C are tangent points of B, D & F are tangent points on G, see diagram).
GF=7=radius of G.
EG=25=distance of E to centre of circle G.
EF
=sqrt(25^2-7^2) (pythagoras)
=sqrt(576)
=24
EA=EC (external tangents from same point E to circle B)
=EF+FA
=24+44
=68
